R. M. Dudley

DISCLAIMER: I only read most of the non-starred sections and did only few of the exercises.

The first half of the book covers real analysis (i.e., topological and metric spaces, measure and integration theory); the second half then introduces basic concepts from probability (mainly the central limit theorem, conditional expectations, and brownian motion) using a lot of the abstract topological stuff from the first part. The advantage of this approach is that many theorems from probability can be stated in a more general form. The disadvantage is that the results become harder to understand. I suspect the author has a tendency towards useless over-generalization of results.

The book is written carefully with much attention to detail, but is often also very terse; I often had to spend 10 minutes to understand a single line. The reason may either be that the author has problems to put himself in the shoes of a person how has not worked on the subject for his entire life, or he wants to "encourage the reader to think"; in both cases it is just odd and a big waste of time, especially since often just a few more words or equations could have reduced the reading time vastly. Another thing that is very annoying, is that the author often changes symbols when there is no need to do so. For example, in a Definition he may call a set "X" only to call the same thing "S" a quarter page later in the theorem that makes use of the Definition.

While the book may be formally self contained, it seems quite inappropriate as a first introduction into the matter. For example, the author often simply seems to have forgotten to state some simple results. Also, I often had problems to look up a certain result and I believe that the reason for this is in mainly this tendency to over-generalize. I conclude from this that while the book contains a huge amount of knowledge one will not be able to make use of that knowledge three month after reading the book.

The examples are purely academic but they often elucidate critical issues of a previous theorem and I found them very helpful. A lot of exercises are provided. The few of them at which I looked seemed to strike a good balance between easy ones and hard ones.… (meer)